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    <meta name="description" content="定义红黑树（Red Black Tree）是一种含有红黑结点并能自平衡的二叉查找树。 它必须除了满足二叉搜索树的性质外，还要满足下面的性质：  每个结点或者为黑色或者为红色。 根结点为黑色。 每个叶结点(实际上就是 NULL 指针)都是黑色的。 如果一个结点是红色的，那么它的两个子节点都是黑色的（也就是说，不能有两个相邻的红色结点）。 对于每个结点，从该结点到其所有子孙叶结点的路径中所包含的黑色结">
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<meta property="og:description" content="定义红黑树（Red Black Tree）是一种含有红黑结点并能自平衡的二叉查找树。 它必须除了满足二叉搜索树的性质外，还要满足下面的性质：  每个结点或者为黑色或者为红色。 根结点为黑色。 每个叶结点(实际上就是 NULL 指针)都是黑色的。 如果一个结点是红色的，那么它的两个子节点都是黑色的（也就是说，不能有两个相邻的红色结点）。 对于每个结点，从该结点到其所有子孙叶结点的路径中所包含的黑色结">
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            <h1 style="display: none">红黑树</h1>
            
            
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                <h1 id="定义"><a href="#定义" class="headerlink" title="定义"></a>定义</h1><p>红黑树（Red Black Tree）是一种含有红黑结点并能自平衡的二叉查找树。</p>
<p>它必须除了满足二叉搜索树的性质外，还要满足下面的性质：</p>
<ol>
<li>每个结点或者为黑色或者为红色。</li>
<li>根结点为黑色。</li>
<li>每个叶结点(实际上就是 NULL 指针)都是黑色的。</li>
<li>如果一个结点是红色的，那么它的两个子节点都是黑色的（也就是说，不能有两个相邻的红色结点）。</li>
<li>对于每个结点，从该结点到其所有子孙叶结点的路径中所包含的黑色结点数量必须相同。</li>
</ol>
<h1 id="变色"><a href="#变色" class="headerlink" title="变色"></a><strong>变色</strong></h1><p>红黑树的变色在 2-3 树中插入节点 3 后，便不再满足 2-3 树的定义，需要进行分解，将元素 2 抽出作为 1 和 3 的父节点，然后 2 继续向上融合。<br>对应到红黑树中就是，首先插入节点 3，在红黑树中新插入的节点默认为红色，然后不满足定义，所以需要进行分解，分解后各个节点都为 2 节点，所以变为黑色。而 2 节点需要继续向上融合，故要变成红色。</p>
<p><img src="/articleimages/%E7%BA%A2%E9%BB%91%E6%A0%91%201f652ec0e17448bbb65292b85ef8e666/Untitled.png" srcset="/img/loading.gif" lazyload alt="Untitled"></p>
<h1 id="右旋"><a href="#右旋" class="headerlink" title="右旋"></a><strong>右旋</strong></h1><p>红黑树的右旋转插入元素 1 后，进行右旋转操作，首先把 2 节点与 3 节点断开连接，同时把 2 与 2 的右子树断开连接，然后把 2 的右子树连接至 3 的左子树位置，不会违背二分搜索树的性质，然后再把 3 连接至 2 的右子树位置。最后还要改变对应节点的颜色，即把 2 节点的颜色改为原来 3 节点的黑色，把 3 节点的颜色改为原来 2 节点的红色。</p>
<p><img src="/articleimages/%E7%BA%A2%E9%BB%91%E6%A0%91%201f652ec0e17448bbb65292b85ef8e666/Untitled%201.png" srcset="/img/loading.gif" lazyload alt="Untitled"></p>
<h1 id="左旋"><a href="#左旋" class="headerlink" title="左旋"></a><strong>左旋</strong></h1><p>与右旋类似，红黑树的左旋转插入元素 3 后，进行左旋转操作，首先把 2 节点与 3 节点断开连接，同时把 3 与 3 的左子树断开连接，然后把 3 的左子树连接至 2 的右子树位置，不会违背二分搜索树的性质，然后再把 2 连接至 3 的左子树位置。最后还要改变对应节点的颜色，即把 2 节点的颜色改为原来 3 节点的红色，把 3 节点的颜色改为原来 2 节点的黑色。</p>
<p><img src="/articleimages/%E7%BA%A2%E9%BB%91%E6%A0%91%201f652ec0e17448bbb65292b85ef8e666/Untitled%202.png" srcset="/img/loading.gif" lazyload alt="Untitled"></p>
<h1 id="示例"><a href="#示例" class="headerlink" title="示例"></a>示例</h1><p>向红黑树中依次插入 10，9，8，7，6，5，4，3，2，1，其如图：</p>
<p><img src="/articleimages/%E7%BA%A2%E9%BB%91%E6%A0%91%201f652ec0e17448bbb65292b85ef8e666/Untitled%203.png" srcset="/img/loading.gif" lazyload alt="Untitled"></p>
<h1 id="实现"><a href="#实现" class="headerlink" title="实现"></a>实现</h1><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span 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class="line">120</span><br><span class="line">121</span><br><span class="line">122</span><br><span class="line">123</span><br></pre></td><td class="code"><pre><code class="hljs java"><span class="hljs-keyword">public</span> <span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">RedBlackTree</span>&lt;<span class="hljs-title">Key</span> <span class="hljs-keyword">extends</span> <span class="hljs-title">Comparable</span>&lt;<span class="hljs-title">Key</span>&gt;, <span class="hljs-title">Value</span>&gt; </span>&#123;<br>    <span class="hljs-keyword">private</span> Node root;      <span class="hljs-comment">//红黑树根结点</span><br>    <span class="hljs-keyword">private</span> <span class="hljs-keyword">int</span> N;  <span class="hljs-comment">//树中元素个数</span><br>    <span class="hljs-keyword">private</span> <span class="hljs-keyword">static</span> <span class="hljs-keyword">final</span> <span class="hljs-keyword">boolean</span> RED = <span class="hljs-keyword">true</span>;        <span class="hljs-comment">//红链接</span><br>    <span class="hljs-keyword">private</span> <span class="hljs-keyword">static</span> <span class="hljs-keyword">final</span> <span class="hljs-keyword">boolean</span> BLACK = <span class="hljs-keyword">false</span>;     <span class="hljs-comment">//黑链接</span><br><br>    <span class="hljs-keyword">private</span> <span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">Node</span></span>&#123;     <span class="hljs-comment">//结点类</span><br>        <span class="hljs-keyword">public</span> Key key;     <span class="hljs-comment">//存储键</span><br>        <span class="hljs-keyword">private</span> Value value;    <span class="hljs-comment">//值</span><br>        <span class="hljs-keyword">public</span> Node left;   <span class="hljs-comment">//左子树</span><br>        <span class="hljs-keyword">public</span> Node right;  <span class="hljs-comment">//右子树</span><br>        <span class="hljs-keyword">public</span> <span class="hljs-keyword">boolean</span> color;   <span class="hljs-comment">//父节点指向它的连接的颜色</span><br><br>        <span class="hljs-function"><span class="hljs-keyword">public</span> <span class="hljs-title">Node</span><span class="hljs-params">(Key key, Value value, Node left, Node right, <span class="hljs-keyword">boolean</span> color)</span> </span>&#123;<br>            <span class="hljs-keyword">this</span>.key = key;<br>            <span class="hljs-keyword">this</span>.value = value;<br>            <span class="hljs-keyword">this</span>.left = left;<br>            <span class="hljs-keyword">this</span>.right = right;<br>            <span class="hljs-keyword">this</span>.color = color;<br>        &#125;<br>    &#125;<br>    <span class="hljs-comment">//获取元素个数</span><br>    <span class="hljs-function"><span class="hljs-keyword">public</span> <span class="hljs-keyword">int</span> <span class="hljs-title">size</span><span class="hljs-params">()</span></span>&#123;<br>        <span class="hljs-keyword">return</span> N;<br>    &#125;<br>    <span class="hljs-comment">//判断连接颜色</span><br>    <span class="hljs-function"><span class="hljs-keyword">public</span> <span class="hljs-keyword">boolean</span> <span class="hljs-title">isRed</span><span class="hljs-params">(Node x)</span></span>&#123;<br>        <span class="hljs-keyword">if</span>(x == <span class="hljs-keyword">null</span>)<br>            <span class="hljs-keyword">return</span> BLACK;<br>        <span class="hljs-keyword">return</span> x.color;<br>    &#125;<br>    <span class="hljs-comment">//左旋转</span><br>    <span class="hljs-function"><span class="hljs-keyword">private</span> Node <span class="hljs-title">rotateLeft</span><span class="hljs-params">(Node h)</span></span>&#123;<br>        <span class="hljs-comment">//获取h结点的右子结点，表示为x</span><br>        Node x = h.right;<br>        <span class="hljs-comment">//让x结点的左子节点称为h结点的右子结点</span><br>        h.right = x.left;<br>        <span class="hljs-comment">//让h称为x结点的左子节点</span><br>        x.left = h;<br>        <span class="hljs-comment">//让x结点的color属性等于h结点的color属性</span><br>        x.color = h.color;<br>        <span class="hljs-comment">//让h结点的color属性变为红色</span><br>        h.color = RED;<br>        <span class="hljs-keyword">return</span> x;<br>    &#125;<br>    <span class="hljs-comment">//右旋</span><br>    <span class="hljs-function"><span class="hljs-keyword">private</span> Node <span class="hljs-title">rotateRight</span><span class="hljs-params">(Node h)</span></span>&#123;<br>        <span class="hljs-comment">//获取h结点的左子结点，表示为x</span><br>        Node x = h.left;<br>        <span class="hljs-comment">//让x结点的右子节点称为h结点的左子结点</span><br>        h.left = x.right;<br>        <span class="hljs-comment">//让h称为x结点的右子节点</span><br>        x.right = h;<br>        <span class="hljs-comment">//让x结点的color属性等于h结点的color属性</span><br>        x.color = h.color;<br>        <span class="hljs-comment">//让h结点的color属性变为红色</span><br>        h.color = RED;<br>        <span class="hljs-keyword">return</span> x;<br>    &#125;<br>    <span class="hljs-comment">//反转颜色</span><br>    <span class="hljs-function"><span class="hljs-keyword">private</span> <span class="hljs-keyword">void</span> <span class="hljs-title">flipColors</span><span class="hljs-params">(Node h)</span></span>&#123;<br>        h.color = RED;<br>        h.left.color = BLACK;<br>        h.right.color = BLACK;<br>    &#125;<br>    <span class="hljs-comment">//插入</span><br>    <span class="hljs-function"><span class="hljs-keyword">public</span> <span class="hljs-keyword">void</span> <span class="hljs-title">put</span><span class="hljs-params">(Key key, Value value)</span></span>&#123;<br>        root = put(root,key,value);<br>        <span class="hljs-comment">//根节点的颜色总是为黑色</span><br>        root.color = BLACK;<br>    &#125;<br>    <span class="hljs-comment">//向指定位置插入</span><br>    <span class="hljs-function"><span class="hljs-keyword">public</span> Node <span class="hljs-title">put</span><span class="hljs-params">(Node h,Key key, Value value)</span></span>&#123;<br>        <span class="hljs-comment">//判断h是否为空，如果为空则直接返回一个红色的结点即可</span><br>        <span class="hljs-keyword">if</span>(h == <span class="hljs-keyword">null</span>)&#123;<br>            N++;<br>            <span class="hljs-keyword">return</span> <span class="hljs-keyword">new</span> Node(key,value,<span class="hljs-keyword">null</span>,<span class="hljs-keyword">null</span>, RED);<br>        &#125;<br>        <span class="hljs-keyword">int</span> cmp = key.compareTo(h.key);<br>        <span class="hljs-keyword">if</span>(cmp&lt;<span class="hljs-number">0</span>)&#123;<br>            <span class="hljs-comment">//继续往左</span><br>            h.left = put(h.left,key,value);<br>        &#125;<span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span>(cmp&gt;<span class="hljs-number">0</span>)&#123;<br>            <span class="hljs-comment">//继续往右</span><br>            h.right = put(h.right,key,value);<br>        &#125;<span class="hljs-keyword">else</span>&#123;<br>            <span class="hljs-comment">//发生值的替换</span><br>            h.value = value;<br>        &#125;<br>        <span class="hljs-comment">//左旋:当前结点的左子节点为黑色，右子节点为红色时，需要左旋</span><br>        <span class="hljs-keyword">if</span>(isRed(h.right) &amp;&amp; !isRed(h.left))&#123;<br>            h = rotateLeft(h);<br>        &#125;<br>        <span class="hljs-comment">//右旋:当前结点的左子节点和左子节点的左子节点都为红色，需要右旋</span><br>        <span class="hljs-keyword">if</span>(isRed(h.left) &amp;&amp; isRed(h.left.left))&#123;<br>            h = rotateRight(h);<br>        &#125;<br>        <span class="hljs-comment">//颜色反转：当前结点的左子结点和右子结点都为红色，需要颜色反转</span><br>        <span class="hljs-keyword">if</span> (isRed(h.left) &amp;&amp; isRed(h.right))&#123;<br>            flipColors(h);<br>        &#125;<br>        <span class="hljs-keyword">return</span> h;<br>    &#125;<br>    <span class="hljs-function"><span class="hljs-keyword">public</span> Value <span class="hljs-title">getKey</span><span class="hljs-params">(Key key)</span></span>&#123;<br><br>        <span class="hljs-keyword">return</span> getKey(root,key);<br>    &#125;<br>    <span class="hljs-function"><span class="hljs-keyword">public</span> Value <span class="hljs-title">getKey</span><span class="hljs-params">(Node x,Key key)</span> </span>&#123;<br>        <span class="hljs-keyword">if</span> (x == <span class="hljs-keyword">null</span>)&#123;<br>            <span class="hljs-keyword">return</span> <span class="hljs-keyword">null</span>;<br>        &#125;<br>        <span class="hljs-comment">//比较x结点的键和key的大小</span><br>        <span class="hljs-keyword">int</span> cmp = key.compareTo(x.key);<br>        <span class="hljs-keyword">if</span>(cmp&lt;<span class="hljs-number">0</span>)&#123;<br>            <span class="hljs-keyword">return</span> getKey(x.left,key);<br>        &#125;<span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span>(cmp&gt;<span class="hljs-number">0</span>)&#123;<br>            <span class="hljs-keyword">return</span> getKey(x.right,key);<br>        &#125;<span class="hljs-keyword">else</span>&#123;<br>            <span class="hljs-keyword">return</span> x.value;<br>        &#125;<br><br>    &#125;<br>&#125;<br></code></pre></td></tr></table></figure>

<h1 id="红黑树在线演示"><a href="#红黑树在线演示" class="headerlink" title="红黑树在线演示"></a>红黑树在线演示</h1><p><a target="_blank" rel="noopener" href="https://www.cs.usfca.edu/~galles/visualization/RedBlack.html">Red/Black Tree</a></p>

                
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